An approximation to the convolution of gamma distributions

In general, the exact distribution of a convolution of independent gamma random variables is quite complicated and does not admit a closed form. Of all the distributions proposed, the gamma-series representation of Moschopoulos (1985 Moschopoulos, P. G. (1985). The distribution of the sum of independent gamma random variables. Annals of the Institute of Statistical Mathematics 37Part A:541–544. [Google Scholar]) is relatively simple to implement but for particular combinations of scale and/or shape parameters the computation of the weights of the series can result in complications with too much time consuming to allow a large-scale application. Recently, a compact random parameter representation of the convolution has been proposed by Vellaisamy and Upadhye (2009 Vellaisamy, P., Upadhye, N. S. (2009). On the sums of compound negative binomial and gamma random variables. Journal of Applied Probability 46:272–283.[Crossref], [Web of Science ®] , [Google Scholar]) and it allows to give an exact interpretation to the weights of the series. They describe an infinite discrete probability distribution. This result suggested to approximate Moschopoulos’s expression looking for an approximating theoretical discrete distribution for the weights of the series. More precisely, we propose a general negative binomial distribution. The result is an “excellent” approximation, fast and simple to implement for any parameter combination.     

Smoothing parameter selection in nonparametric regression using an improved Akaike information criterion

Many different methods have been proposed to construct nonparametric estimates of a smooth regression function, including local polynomial, (convolution) kernel and smoothing spline estimators. Each of these estimators uses a smoothing parameter to control the amount of smoothing performed on a given data set. In this paper an improved version of a criterion based on the Akaike information criterion (AIC), termed AIC_C, is derived and examined as a way to choose the smoothing parameter. Unlike plug-in methods, AIC_C can be used to choose smoothing parameters for any linear smoother, including local quadratic and smoothing spline estimators. The use of AIC_C avoids the large variability and tendency to undersmooth (compared with the actual minimizer of average squared error) seen when other 'classical' approaches (such as generalized cross-validation (GCV) or the AIC) are used to choose the smoothing parameter. Monte Carlo simulations demonstrate that the AIC_C-based smoothing parameter is competitive with a plug-in method (assuming that one exists) when the plug-in method works well but also performs well when the plug-in approach fails or is unavailable.     

The Lindeberg-Feller theorem for positive definite densities

We consider the Lindeberg-Feller model for independent random variables and focus our attention on the behaviour of the probability densities q_{n} of sums S_{n}, n\geq 1 . We obtain a theorem on the convergence of q_{n} to the standard normal density \varphi which resembles the well known limit theorem for distribution functions--provided that the q_{n} are positive definite. A special case is the following: if q_{n}(0)\rightarrow\varphi(0) as n\rightarrow\infty then the Lindeberg condition guarantees that the convergence of q_{n} to \varphi continues to the real line.     

Convergence of non-linear functionals of smoothed empirical processes and kernel density estimates

We consider regularizations by convolution of the empirical process and study the asymptotic behaviour of non-linear functionals of this process. Using a result for the same type of non-linear functionals of the Brownian bridge, shown in a previous paper [4], and a strong approximation theorem, we prove several results for the p-deviation in estimation of the derivatives of the density. We also study the asymptotic behaviour of the number of crossings of the smoothed empirical process defined by Yukich [17] and of a modified version of the Kullback deviation.     

The tail behavior of the convolutions of Gamma random variables

Two sufficient conditions for comparing convolutions of heterogeneous gamma random variables in terms of star order are established. It is further shown that if the scale parameters of heterogeneous gamma random variables are more dispersed in the sense of majorization, then the convolutions are more dispersed according to the right spread order, which generalizes and strengthens the results in Diaconis and Perlman (1987), Kochar and Xu (2010) and Zhao and Balakrishnan (2009).     

A non‐linear model of fecundability,postpartum amenorrhea,and sterility*

This paper proposes a convolution model of fecundability, controling for the effects of postpartum amenorrhea and unobserved heterogeneity in fecundability. Simulation analysis was used to assess the validity and reliability of estimates derived from the model. Analysis showed that the model captured the mean and standard deviation of age at the onset of sterility in simulated populations where sterility followed either a Gompertz, a gamma, or a lognormal distribution. The model performed well when sterility was specified by either a lognormal or a gamma distribution. The model also accurately estimated fecundability and postpartum amenorrhea. Next, the model was found to fit data from 17th and 18th century French Canadian birth histories. In this French Canadian sample the mean age at sterility was found to be 46.3 years using a gamma model. The decline in fecundability was almost linear after age 30. Thus, fecundability at age 40 had declined to about one‐third of that observed at age 30. Variability in individual fecundability was quite high. For example, women with fecundability one standard deviation above the mean had about 2.3 times as high fecundability as women one standard deviation below the mean.     

A model of the age patterns of births by parity in natural fertility populations

This research develops a convolution model to express the age patterns of fertility at each birth order in natural fertility populations in terms of six parameters, directly representing the proximate determinants of fertility, and a series of parity level indicators. The parity level indicators at each birth order are simply the proportions of women in a cohort who will eventually have births at each birth order it the age‐related fecundity decline is controlled. The Coale‐McNeil nuptiality model is adopted to represent the age pattern of first marriage rates and the natural fertility schedule employed in the Coale‐Trussell fertility model is incorporated to adjust age effects. The fast Fourier transform is used in solving the model numerically. It proves that the model is able to provide excellent fits to fertility for rural Chinese women in the 1950s.     

Instructional Uses of Approximate Convolutions and Their Graphs

A technique is presented for approximating convolution densities f * g by forming the convolution of step-function approximations of f and g. This technique, amenable to the electronic calculator or computer, provides suitably accurate graphs for illustrating concepts in probability and statistics. Several examples are included.     

Some properties of discrete bathtub-shaped distributions

The existence and the usefulness of discrete bathtub-shaped and upside down bathtub-shaped distributions have been demonstrated in some papers of recent origin. However, the general properties of these two classes of distributions do not seem to have been discussed. This article proposes to study some reliability properties of such distributions. We investigate the closure properties with reference to convolution, mixing, series and parallel systems, etc. and existence of bounds on reliability functions, moment properties, and convergence.